Large-data global existence in a higher-dimensional doubly degenerate nutrient system

Abstract

This paper deals with a doubly degenerate nutrient taxis system given by(⋆){ut=∇⋅(uv∇u)−χ∇⋅(uαv∇v)+ℓuv,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, in a smoothly bounded convex domain Ω⊂Rn, n=2,3, with zero-flux boundary conditions, where α>0,χ>0 and ℓ≥0.It is known that when α=2, for which, in fact, the system was proposed by Leyva et al. (2013) to model the aggregation patterns of colonies of Bacillus subtilis identified on the surface of thin agar plates, a globally defined weak solution to (⋆) can be obtained, either in the case n=1, or for its corresponding spatially two-dimensional version, the latter case additionally requiring certain restrictions on the size of initial data. The present paper complements the results, asserting that under the assumptionα∈{(1,32)if n=2,and(76,139)if n=3, for all sufficiently smooth initial data, (⋆) possesses such a global weak solution (u,v).